To extend A018216 Maximal number of subgroups in a group with n elements

[Christian G. Bower]

so this is a lower bound if the order is a power of 2.
Return-Path: <maxale at gmail.com>
X-Ids: 164
DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed;
        d=gmail.com; s=gamma;
        h=domainkey-signature:received:received:message-id:date:from:to:subject:cc:in-reply-to:mime-version:content-type:content-transfer-encoding:content-disposition:references;
        bh=ylzdKnnWsUOFtRBKhWKAVB39QCCEJgFpfC3wVJSThfc=;
        b=qjoUsFhX0Qh/etmLugB1PeoLyvsoNZIOyO9wvrOwCpCfO7sU49tfH2bq+ufdHjnmUuW4l6zkawLtZ/YSCiubOWZSnzmJ1yTRAuxv/r3AlUtUOH1/p6AlPa24MdSuTHJSgnlHKK5EoGi2vWHZh3nxFP5OCKhWgnGu3ufbXxh5dH8=
DomainKey-Signature: a=rsa-sha1; c=nofws;
        d=gmail.com; s=gamma;
        h=received:message-id:date:from:to:subject:cc:in-reply-to:mime-version:content-type:content-transfer-encoding:content-disposition:references;
        b=JIOge4GHTWvsGOpFi2eUvICgs0H86Dnc91Ds4ufLk3GqfmxZuKahKk5YTwOs4dHyitGlsHUl0EnwuqY9B1JIn71GeFMNSWVNpGAnUbcnLKsCkvzuE9q4Xh9DRC0j9hm+kyArZlQx4zIVp90ywP8L9y0GIDn4b1+PiLu7ghpMj1I=
Message-ID: <d3dac270712011727p3ce0a022if2e1fb5e1825c2f5 at mail.gmail.com>
Date: Sat, 1 Dec 2007 17:27:44 -0800
From: "Max Alekseyev" <maxale at gmail.com>
To: "Christian G. Bower" <bowerc at usa.net>
Subject: Re: To extend A018216 Maximal number of subgroups in a group with n elements
Cc: seqfan <seqfan at ext.jussieu.fr>, "N. J. A. Sloane" <njas at research.att.com>
In-Reply-To: <586LLaaON9612S02.1196469673 at cmsweb02.cms.usa.net>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 7bit
Content-Disposition: inline
References: <586LLaaON9612S02.1196469673 at cmsweb02.cms.usa.net>
X-Greylist: IP, sender and recipient auto-whitelisted, not delayed by milter-greylist-3.0 (shiva.jussieu.fr [134.157.0.164]); Sun, 02 Dec 2007 02:27:46 +0100 (CET)
X-Virus-Scanned: ClamAV 0.88.7/4970/Sat Dec  1 16:44:03 2007 on shiva.jussieu.fr
X-Virus-Status: Clean
X-j-chkmail-Score: MSGID : 47520A12.000 on shiva.jussieu.fr : j-chkmail score : X : 0/50 1 0.321 -> 1
X-Miltered: at shiva.jussieu.fr with ID 47520A12.000 by Joe's j-chkmail (http://j-chkmail.ensmp.fr)!

On Nov 30, 2007 4:41 PM, Christian G. Bower <bowerc at usa.net> wrote:
> I think C2^4 has 67 subgroups (1 trivial, 15 C2, 35 C2^2, 15 C2^3, 1 itself).
> I would suspect that's the largest case, but I'm not in the mood to check all
> 14 groups of order 16 (and to dig up a description of the more obsure ones.)

I've checked all abelian groups of order 16 and A006116(4)=67 is
indeed the largest case:

C16 has 5 subgroups
C2 x C8 has 11 subgroups
(C2)^2 x C4 has 27 subgroups
(C2)^4 has 67 subgroups
(C4)^2 has 15 subgroups

This gives boost to A061034:

%S A061034 1,2,2,5,2,4,2,16,6,4,2,10,2,4,4,67,2,12,2,10,4,4,2,32,8,4,28,10,2,8,2

Neil, please update A061034 accordingly.

Regards,
Max